Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T22:21:00.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Zeno breaking, the Contact effect and sensitive behaviour in piecewise-linear systems

Published online by Cambridge University Press:  21 March 2018

R. EDWARDS
R. EDWARDS*
Affiliation:
Department of Mathematics, University of Victoria, PO Box 1700 STN CSC, Victoria, BC V8W 2Y2, Canada email: edwards@uvic.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Non-smooth approximations of steep sigmoidal switching networks, such as those used as qualitative models of gene regulation, lead to analytic and computational challenges that arise as a result of the discontinuities in the vector fields. In order to highlight the need for care in dealing with such systems, several particular phenomena are presented here through illustrative examples, including ‘Zeno breaking’, or computing beyond the finite time convergence of an infinite sequence of threshold transitions; the ‘Contact’ effect, in which in the discontinuous limit, trajectories can pass through a ‘saddle point’ without stopping, though these solutions are not unique and other solutions stop for arbitrary time intervals; and sensitive behaviour that arises from exotic dynamics within switching regions.

Keywords

Discontinuous equationssingular perturbationsdifferential inclusionsoscillationsdynamical systems in biology (34A36; 34D15; 34A60; 34C10; 37N25)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Special Issue 5: Theory and applications of nonsmooth dynamical systems , October 2018 , pp. 826 - 844
Copyright
Copyright © Cambridge University Press 2018 

Footnotes

†This work was partially supported by a Discovery Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada.

References

[1] Acary, V., de Jong, H. & Brogliato, B. (2014) Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems. Physica D 269, 103–119.Google Scholar
[2] Ames, A. D., Zheng, H., Gregg, R. D. & Sastry, S. (2006) Is there life after Zeno? Taking executions past the breaking (Zeno) point. In: Proceedings of the 2006 American Control Conference, 14–16 June 2006, Minneapolis, Minnesota: IEEE, 2652–2657.Google Scholar
[3] Artstein, Z. (2002) On singularly perturbed ordinary differential equations with measure-valued limits. Math. Bohemica 127 (2), 139–152.Google Scholar
[4] Artstein, Z., Linshiz, J. & Titi, E. S. (2007) Young measure approach to computing slowly advancing fast oscillations. Multiscale Model. Simul. 6 (4), 1085–1097.Google Scholar
[5] Artstein, Z. & Vigodner, A. (1996) Singularly perturbed ordinary differential equations with dynamic limits. Proc. R. Soc. Edinburgh: Sect. A Math. 126 (3), 541–569.Google Scholar
[6] Dieci, L. & Difonzo, F. (2014) A comparison of Filippov sliding vector fields in codimension 2. J. Comput. Appl. Math. 262, 161–179.Google Scholar
[7] Del Buono, N., Elia, C. & Lopez, L. (2014) On the equivalence between the sigmoidal approach and Utkin's approach for piecewise-linear models of gene regulatory networks. SIAM J. Appl. Dyn. Syst. 13 (3), 1270–1292.Google Scholar
[8] Edwards, R. (2000) Analysis of continuous-time switching networks. Physica D 146, 165–199.Google Scholar
[9] Edwards, R., Hill, A. & Jacquier, M. (2014) Analysis of transient damped oscillations in gene regulatory networks. In: Proceedings of the MTNS Conference, July 2014, Groningen, the Netherlands.Google Scholar
[10] Filippov, A. F. (1988) Differential Equations with Discontinuous Righthand Sides, Kluwer, Dordrecht.Google Scholar
[11] Gouzé, J.-L. & Sari, T. (2002) A class of piecewise linear differential equations arising in biological models. Dyn. Syst. 17 (4), 299–316.Google Scholar
[12] Guglielmi, N. & Hairer, E. (2015) Classification of hidden dynamics in discontinuous dynamical systems. SIAM J. Appl. Dyn. Syst. 14 (3), 1454–1477.Google Scholar
[13] Hudson, D. & Edwards, R. (2016) Dynamics of transcription-translation networks. Physica D 331, 102–113.Google Scholar
[14] Ironi, L., Panzeri, L., Plahte, E. & Simoncini, V. (2011) Dynamics of actively regulated gene networks. Physica D 240 (8), 779–794.Google Scholar
[15] Ironi, L. & Tran, D. X. (2016) Nonlinear and temporal multiscale dynamics of gene regulatory networks: A qualitative simulator. Math. Comput. Simul. 125, 15–37.Google Scholar
[16] Jeffrey, M. R. (2014) Dynamics at a switching intersection: Hierarchy, isonomy, and multiple sliding. SIAM J. Appl. Dyn. Syst. 13 (3), 1082–1105.Google Scholar
[17] Machina, A., Edwards, R. & van den Driessche, P. (2013) Singular dynamics in gene network models. SIAM J. Appl. Dyn. Syst. 12 (1), 95–125.Google Scholar
[18] Machina, A., Edwards, R. & van den Driessche, P. (2013) Sensitive dependence on initial conditions in gene networks. Chaos 23 (2), 025101.Google Scholar
[19] Machina, A. & Ponosov, A. (2011) Filippov solutions in the analysis of piecewise linear models describing gene regulatory networks. Nonlinear Anal. 74 (3), 882–900.Google Scholar
[20] Mestl, T., Plahte, E. & Omholt, S. W. (1995) Periodic solutions in systems of piecewise-linear differential equations. Dyn. Stab. Syst. 10 (2), 179–193.Google Scholar
[21] Plahte, E. & Kjøglum, S. (2005) Analysis and generic properties of gene regulatory networks with graded response functions. Physica D 201 (1–2), 150–176.Google Scholar
[22] Verhulst, F. (1996) Nonlinear Differential Equations and Dynamical Systems, 2nd ed., Springer, Berlin.Google Scholar
[23] Webber, S., Glendinning, P. & Jeffrey, M. R. (2018) Pausing in piecewise-smooth dynamic systems, in preparation.Google Scholar